partial trace

Suppose $V$, $W$ are finite-dimensional vector spaces over a field, with dimensions $m$ and $n$, respectively. For any space $A$ let $L(A)$ denote the space of linear operators on $A$. The partial trace over $W$, Tr$_{W}$, is a mapping

$T \in L(V \otimes W) \mapsto Tr_{W}(T) \in L(V).$

Let $e_{1}, \ldots, e_{m}$ and $f_{1}, \ldots, f_{n}$ be bases for $V$ and $W$ respectively. Then $T$ has a matrix representation $\{a_{kl,ij}\}$ where $1 \le k,i \le m$ and $1 \le l,j \le n$ relative to the basis of the space $V \otimes W$ given by $e_{k} \otimes f_{l}$. Consider the sum

$b_{k,i} = \sum_{j=1}^{n}a_{kj,ij}$

for $k,i$ over $1, \ldots, m$. This gives the matrix $b_{k,i}$. The associated linear operator on $V$ is independent of the choice of bases and is defined as the partial trace.

Consider a quantum system, $\rho$, in the presence of an environment, $\rho_{env}$. Consider what is known in quantum information theory as the CNOT gate:

$U={|00\rangle}{\langle 00|} + {|01\rangle}{\langle 01|} + {|11\rangle}{\langle 10|} + {|10\rangle}{\langle 11|}.$

Suppose our system has the simple state ${|1\rangle}{\langle 1|}$ and the environment has the simple state ${|0\rangle}{\langle 0|}$. Then $\rho \otimes \rho_{env} = {|10\rangle}{\langle 10|}$. In the quantum operation formalism we have

$T(\rho) = \frac{1}{2}Tr_{env}U(\rho \otimes \rho_{env})U^{\dagger}
= \frac{1}{2}Tr_{env}({|10\rangle}{\langle 10|} + {|11\rangle}{\langle 11|})
= \frac{{|1\rangle}{\langle 1|}{\langle 0|0\rangle} + {|1\rangle}{\langle 1|}{\langle 1|1\rangle}}{2}
= {|1\rangle}{\langle 1|}$

where we inserted the normalization factor $\frac{1}{2}$.

Revised on March 6, 2010 23:40:54
by Toby Bartels
(75.117.106.207)